Idols are a concept that don't really exist in the West, and even the idols in other Asian countries such as South Korea are quite different from that of Japan. But to some it up quickly: idols are a type of celebrity in Japan that do a bit of everything. From singing, to dancing, to acting; they may not be the best at all of those things combined but their popularity generally comes more from who they are and their image rather than their actual work. They are usually in girl groups or boybands (girl groups being the most popular) but there are also solo idols and mixed groups. For female idols in particular they tend to be fairly young (under 25) and have a specific image of purity to adhere to, although there are many groups nowadays who break these “rules”. -Dansei Idol Girl
An Idol is anyone who says they are an Idol and has fans that support them like an idol. That was an unsatisfying answer.
Unfortunately giving a definition of an Idol is tricky. The best thing I can say is, you know it when you see it (or hear it). There are a bunch of concepts that are factors for how Idolish something is. The more concepts like, Cute, Young, Produced, Personal Development, Ouedan (Fan Support), Singing, Dancing, Joyful, Ballerina Aesthetic, Doll Aesthetic, an act has, the more likely it is to be an Idol. It used to be a formula, but now its just basically, if you identify as an idol, you are an idol.
Wikipedia says that an idol is a young starlet manufactured and marketed for image, attractiveness, and personality in Japanese pop culture. It says that they are primarily singers, but also engage in activities, such as acting, modeling, and appearing in variety shows. This is mostly true for Chijou Idols, but not as much forChika Idols. There is a really diverse amount of experiences as an Idol and an Idol fan.
Wikipedia says that an idol's main objective is to entertain and offer an escapism from daily life. This is the the most unwavering, unifying, and absolute characteristic of Idol Culture. The goal of Idols is to provide entertainment and offer an escape from life.
So there are many levels in an endeavor. You have Professionals, Semi-Professionals, Enthusiasts, Hobbyists, and Amatuers. Most Idols are around the Hobbyist level of skill in most of the things they do (otherwise they would be doing what they do on a semi professional level or higher). Most Idols can't sing very well, or dance very well. But neither can their fans. So unlike a professional Enka singer's fan who go to Karaoke, Idol Fans can sing and dance along with most Idol Songs. Most people can not sing along with Enka unless they can sing pretty well. Most people can not cover a Ballerina's dance unless they can dance pretty well.
So if Idols can not sing or dance well, why do people like them and support them? Simply put, because idols are cute (or cool in the case of male idols). Idols create a cult of personality around themselves and amass a group of loyal followers with their looks and charms. Using the media to reach out to more and more people, their group of followers gets bigger and bigger, until a company feels that they can get money by associating themselves with that Idol. Then the company starts paying the Talent company money for the Idol to do jobs for them. (Hey take pictures in my clothes, Hey do an advertisement with my product, hey come talk on my show, hey come act in my tv show, hey let me use your hit song for my movie, etc…).
I think initially the Idol Culture was started to sell the music of boring old guys. Hey lets get this cute girl to sing songs with lyrics a Middle Aged man would love to hear with the latest song I wrote. But then the industry found other ways to use idols.
Idols are not Professional Singers. Some Idols become Professional Singers after they graduate from being Idols, but there is virtually 0% overlap between being a Professional Singer and Being an Idol.
Talent agencies commercialize idols by recruiting boys and girls with little or no prior experience in the entertainment industry, often as aspiring stars/starlets with the intent of creating a passionate following. Many fans of idols see them as akin to siblings or girl/boy next door types and empathize with the idols and enthusiastically follow their growth from ordinary, inexperienced amateurs to famous, experienced artists.
Check out this link about Aidoru Pops.